Chapter 4. Clustering and classification

Analysis part

Boston data has 506 observations and 14 variables. It has information about housing in suburbs of Boston.

Description of variables in the Boston dataset:

  1. crim - per capita crime rate by town.

  2. zn - proportion of residential land zoned for lots over 25,000 sq.ft.

  3. indus - proportion of non-retail business acres per town.

  4. chas - Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).

  5. nox - nitrogen oxides concentration (parts per 10 million).

  6. rm - average number of rooms per dwelling.

  7. age - proportion of owner-occupied units built prior to 1940.

  8. dis - weighted mean of distances to five Boston employment centres.

  9. rad - index of accessibility to radial highways.

  10. tax - full-value property-tax rate per $10,000.

  11. ptratio - pupil-teacher ratio by town.

  12. black - 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town.

  13. lstat - lower status of the population (percent).

  14. medv - median value of owner-occupied homes in $1000s.

2. Load Boston data from MASS package

# Load required libraries
library(ggplot2)
library(dplyr)
library(corrplot)
library(GGally)
library(tidyr)
library(MASS)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
data('Boston')

str(Boston)
## 'data.frame':    506 obs. of  14 variables:
##  $ crim   : num  0.00632 0.02731 0.02729 0.03237 0.06905 ...
##  $ zn     : num  18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
##  $ indus  : num  2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
##  $ chas   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ nox    : num  0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
##  $ rm     : num  6.58 6.42 7.18 7 7.15 ...
##  $ age    : num  65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
##  $ dis    : num  4.09 4.97 4.97 6.06 6.06 ...
##  $ rad    : int  1 2 2 3 3 3 5 5 5 5 ...
##  $ tax    : num  296 242 242 222 222 222 311 311 311 311 ...
##  $ ptratio: num  15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
##  $ black  : num  397 397 393 395 397 ...
##  $ lstat  : num  4.98 9.14 4.03 2.94 5.33 ...
##  $ medv   : num  24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
dim(Boston)
## [1] 506  14
summary(Boston)
##       crim                zn             indus            chas        
##  Min.   : 0.00632   Min.   :  0.00   Min.   : 0.46   Min.   :0.00000  
##  1st Qu.: 0.08204   1st Qu.:  0.00   1st Qu.: 5.19   1st Qu.:0.00000  
##  Median : 0.25651   Median :  0.00   Median : 9.69   Median :0.00000  
##  Mean   : 3.61352   Mean   : 11.36   Mean   :11.14   Mean   :0.06917  
##  3rd Qu.: 3.67708   3rd Qu.: 12.50   3rd Qu.:18.10   3rd Qu.:0.00000  
##  Max.   :88.97620   Max.   :100.00   Max.   :27.74   Max.   :1.00000  
##       nox               rm             age              dis        
##  Min.   :0.3850   Min.   :3.561   Min.   :  2.90   Min.   : 1.130  
##  1st Qu.:0.4490   1st Qu.:5.886   1st Qu.: 45.02   1st Qu.: 2.100  
##  Median :0.5380   Median :6.208   Median : 77.50   Median : 3.207  
##  Mean   :0.5547   Mean   :6.285   Mean   : 68.57   Mean   : 3.795  
##  3rd Qu.:0.6240   3rd Qu.:6.623   3rd Qu.: 94.08   3rd Qu.: 5.188  
##  Max.   :0.8710   Max.   :8.780   Max.   :100.00   Max.   :12.127  
##       rad              tax           ptratio          black       
##  Min.   : 1.000   Min.   :187.0   Min.   :12.60   Min.   :  0.32  
##  1st Qu.: 4.000   1st Qu.:279.0   1st Qu.:17.40   1st Qu.:375.38  
##  Median : 5.000   Median :330.0   Median :19.05   Median :391.44  
##  Mean   : 9.549   Mean   :408.2   Mean   :18.46   Mean   :356.67  
##  3rd Qu.:24.000   3rd Qu.:666.0   3rd Qu.:20.20   3rd Qu.:396.23  
##  Max.   :24.000   Max.   :711.0   Max.   :22.00   Max.   :396.90  
##      lstat            medv      
##  Min.   : 1.73   Min.   : 5.00  
##  1st Qu.: 6.95   1st Qu.:17.02  
##  Median :11.36   Median :21.20  
##  Mean   :12.65   Mean   :22.53  
##  3rd Qu.:16.95   3rd Qu.:25.00  
##  Max.   :37.97   Max.   :50.00

3. Graphical overview of the data.

pairs(Boston)

# print the correlation matrix
cor_matrix<-cor(Boston) 
cor_matrix %>% round(digits = 2)
##          crim    zn indus  chas   nox    rm   age   dis   rad   tax
## crim     1.00 -0.20  0.41 -0.06  0.42 -0.22  0.35 -0.38  0.63  0.58
## zn      -0.20  1.00 -0.53 -0.04 -0.52  0.31 -0.57  0.66 -0.31 -0.31
## indus    0.41 -0.53  1.00  0.06  0.76 -0.39  0.64 -0.71  0.60  0.72
## chas    -0.06 -0.04  0.06  1.00  0.09  0.09  0.09 -0.10 -0.01 -0.04
## nox      0.42 -0.52  0.76  0.09  1.00 -0.30  0.73 -0.77  0.61  0.67
## rm      -0.22  0.31 -0.39  0.09 -0.30  1.00 -0.24  0.21 -0.21 -0.29
## age      0.35 -0.57  0.64  0.09  0.73 -0.24  1.00 -0.75  0.46  0.51
## dis     -0.38  0.66 -0.71 -0.10 -0.77  0.21 -0.75  1.00 -0.49 -0.53
## rad      0.63 -0.31  0.60 -0.01  0.61 -0.21  0.46 -0.49  1.00  0.91
## tax      0.58 -0.31  0.72 -0.04  0.67 -0.29  0.51 -0.53  0.91  1.00
## ptratio  0.29 -0.39  0.38 -0.12  0.19 -0.36  0.26 -0.23  0.46  0.46
## black   -0.39  0.18 -0.36  0.05 -0.38  0.13 -0.27  0.29 -0.44 -0.44
## lstat    0.46 -0.41  0.60 -0.05  0.59 -0.61  0.60 -0.50  0.49  0.54
## medv    -0.39  0.36 -0.48  0.18 -0.43  0.70 -0.38  0.25 -0.38 -0.47
##         ptratio black lstat  medv
## crim       0.29 -0.39  0.46 -0.39
## zn        -0.39  0.18 -0.41  0.36
## indus      0.38 -0.36  0.60 -0.48
## chas      -0.12  0.05 -0.05  0.18
## nox        0.19 -0.38  0.59 -0.43
## rm        -0.36  0.13 -0.61  0.70
## age        0.26 -0.27  0.60 -0.38
## dis       -0.23  0.29 -0.50  0.25
## rad        0.46 -0.44  0.49 -0.38
## tax        0.46 -0.44  0.54 -0.47
## ptratio    1.00 -0.18  0.37 -0.51
## black     -0.18  1.00 -0.37  0.33
## lstat      0.37 -0.37  1.00 -0.74
## medv      -0.51  0.33 -0.74  1.00
# Histograms of the variables
Boston %>% 
  gather(key=var_name, value = value) %>% 
  ggplot(aes(x=value)) +
  geom_histogram() +
  facet_wrap(~var_name, scales = "free_x")
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

# Create color ramp from dark blue to white to red.
colorVector <- c("blue", "white", "red")

# visualize the correlation matrix
corrplot(cor_matrix, method="circle", type = "upper", cl.pos = "b", tl.pos = "d", tl.cex = 0.6, col = colorRampPalette(colorVector)(200))

The histogram plot shows that much of the data does not look like gaussian normally distributed data. Most of the data variables has a high kurtosis, or has two peaks. The correlation plot shows that there are many high positive correlations between different variables: like industy and NO2 gas level and tax revenue; property taxes and access to radial highways. There are some strong negative correlations ones also like median value of owner-occupied homes and lower status of the population; and between distances to five Boston employment centres and proportion of owner-occupied units built prior to 1940.

4. Standardize the dataset. Create crime rate categorical variable. Split data into training and data parts.

# center and standardize variables
boston_scaled <- scale(Boston)

# summaries of the scaled variables
summary(boston_scaled)
##       crim                 zn               indus        
##  Min.   :-0.419367   Min.   :-0.48724   Min.   :-1.5563  
##  1st Qu.:-0.410563   1st Qu.:-0.48724   1st Qu.:-0.8668  
##  Median :-0.390280   Median :-0.48724   Median :-0.2109  
##  Mean   : 0.000000   Mean   : 0.00000   Mean   : 0.0000  
##  3rd Qu.: 0.007389   3rd Qu.: 0.04872   3rd Qu.: 1.0150  
##  Max.   : 9.924110   Max.   : 3.80047   Max.   : 2.4202  
##       chas              nox                rm               age         
##  Min.   :-0.2723   Min.   :-1.4644   Min.   :-3.8764   Min.   :-2.3331  
##  1st Qu.:-0.2723   1st Qu.:-0.9121   1st Qu.:-0.5681   1st Qu.:-0.8366  
##  Median :-0.2723   Median :-0.1441   Median :-0.1084   Median : 0.3171  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.:-0.2723   3rd Qu.: 0.5981   3rd Qu.: 0.4823   3rd Qu.: 0.9059  
##  Max.   : 3.6648   Max.   : 2.7296   Max.   : 3.5515   Max.   : 1.1164  
##       dis               rad               tax             ptratio       
##  Min.   :-1.2658   Min.   :-0.9819   Min.   :-1.3127   Min.   :-2.7047  
##  1st Qu.:-0.8049   1st Qu.:-0.6373   1st Qu.:-0.7668   1st Qu.:-0.4876  
##  Median :-0.2790   Median :-0.5225   Median :-0.4642   Median : 0.2746  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.6617   3rd Qu.: 1.6596   3rd Qu.: 1.5294   3rd Qu.: 0.8058  
##  Max.   : 3.9566   Max.   : 1.6596   Max.   : 1.7964   Max.   : 1.6372  
##      black             lstat              medv        
##  Min.   :-3.9033   Min.   :-1.5296   Min.   :-1.9063  
##  1st Qu.: 0.2049   1st Qu.:-0.7986   1st Qu.:-0.5989  
##  Median : 0.3808   Median :-0.1811   Median :-0.1449  
##  Mean   : 0.0000   Mean   : 0.0000   Mean   : 0.0000  
##  3rd Qu.: 0.4332   3rd Qu.: 0.6024   3rd Qu.: 0.2683  
##  Max.   : 0.4406   Max.   : 3.5453   Max.   : 2.9865
# change the object to data frame from matrix type.
boston_scaled <- as.data.frame(boston_scaled)

sd(boston_scaled$crim)
## [1] 1
# Now all the variables have zero mean and SD of 1. This is recommended for clustering. 


# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins
##           0%          25%          50%          75%         100% 
## -0.419366929 -0.410563278 -0.390280295  0.007389247  9.924109610
# create a categorical variable 'crime'
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label = c("low", "med_low", "med_high", "high"))

# look at the table of the new factor crime. The bins have either 127 or 126 elements.
table(crime)
## crime
##      low  med_low med_high     high 
##      127      126      126      127
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)

# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)


# Select 80 % of the data and 20 % of the data and split them into separate datasets.
# number of rows in the Boston dataset 
n <- nrow(Boston)

# choose randomly 80% of the rows
ind <- sample(n,  size = n * 0.8)

# create training set
train <- boston_scaled[ind,]

# create test set 
test <- boston_scaled[-ind,]

# save the correct classes from test data
correct_classes <- test$crime

# remove the crime variable from test data
test <- dplyr::select(test, -crime)

5. Fit linear discriminant analysis on the training set.

# linear discriminant analysis. The . means all of the variables.
lda.fit <- lda(crime ~ ., data = train)

# print the lda.fit object. Note that the first linear discriminant explains 95 % of the model, and the third one only 1.2 %.
lda.fit
## Call:
## lda(crime ~ ., data = train)
## 
## Prior probabilities of groups:
##       low   med_low  med_high      high 
## 0.2301980 0.2623762 0.2450495 0.2623762 
## 
## Group means:
##                   zn      indus        chas        nox         rm
## low       0.91525712 -0.8908605 -0.10299142 -0.8483863  0.4295002
## med_low  -0.08334565 -0.2951311 -0.08661679 -0.5872960 -0.1323387
## med_high -0.39109146  0.1689534  0.28443258  0.3830397  0.1095356
## high     -0.48724019  1.0170298 -0.12375925  1.0380435 -0.4033470
##                 age        dis        rad        tax     ptratio
## low      -0.8271561  0.8408270 -0.7015464 -0.7233055 -0.43441270
## med_low  -0.3418616  0.4098612 -0.5495709 -0.4993659 -0.06704845
## med_high  0.4349117 -0.3920332 -0.4470800 -0.3546549 -0.34478590
## high      0.8146248 -0.8527739  1.6390172  1.5146914  0.78181164
##               black        lstat         medv
## low       0.3819121 -0.757191246  0.497525719
## med_low   0.3125279 -0.151112125 -0.002233548
## med_high  0.0562525 -0.006157674  0.189950189
## high     -0.7998711  0.928770666 -0.702925805
## 
## Coefficients of linear discriminants:
##                  LD1         LD2         LD3
## zn       0.091597782  0.72582560 -1.08552595
## indus    0.075032287 -0.20080401  0.50705166
## chas    -0.136066170 -0.11949253 -0.05054510
## nox      0.373836580 -0.84617639 -1.24604452
## rm      -0.153705564 -0.11732365 -0.17308915
## age      0.062845066 -0.30311814 -0.04593389
## dis     -0.134670630 -0.27855764  0.44857222
## rad      3.728089299  0.81545803  0.06238177
## tax     -0.007431938  0.08332912  0.32549723
## ptratio  0.108544816  0.01511329 -0.33385861
## black   -0.091260534  0.06675804  0.11447525
## lstat    0.243331995 -0.22221458  0.32180038
## medv     0.202650958 -0.44362332 -0.10349837
## 
## Proportion of trace:
##    LD1    LD2    LD3 
## 0.9606 0.0291 0.0104
# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
  heads <- coef(x)
  arrows(x0 = 0, y0 = 0, 
         x1 = myscale * heads[,choices[1]], 
         y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
  text(myscale * heads[,choices], labels = row.names(heads), 
       cex = tex, col=color, pos=3)
}

# target classes as numeric
classes <- as.numeric(train$crime)

# plot the lda results. Both lines need to be run at the same time. High class of crime rate looks to cluster pretty well with only a few medium high class elements in it, and only one high class element being at LD1 value of ~0.
plot(lda.fit, col = classes, pch = classes, dimen = 2)
lda.arrows(lda.fit, myscale = 1)

6. Predict the classes with the LDA model.

# Saving of the crime variable in a vector was done in step 5. Remove the crime variable from test data not necessary either.
# test <- dplyr::select(test, -crime)

# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)

# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)
##           predicted
## correct    low med_low med_high high
##   low       22      12        0    0
##   med_low    4      13        3    0
##   med_high   0       8       16    3
##   high       0       0        0   21

The categorization of the low and especially high classes of properties was quite successful. The medium low and medium high classes were only about 60 % correct. There was more confusion about the low crime rate properties compared to the high crime rating properties. Having shifting values depending on different runs of the script is annoing.

7. Start again with the Boston dataset. Test what number of clusters to run K-means clustering with would make sense.

# Reload Boston dataset.
data("Boston")

# center and standardize variables
boston_scaled <- scale(Boston)

# change the object to data frame from matrix type.
boston_scaled <- as.data.frame(boston_scaled)

# Calculate the Euclidean distances between observations.
dist_eu <- dist(boston_scaled)

# look at the summary of the distances
summary(dist_eu)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.1343  3.4625  4.8241  4.9111  6.1863 14.3970
# K-means clustering
km <-kmeans(boston_scaled, centers = 3)

# plot the Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)

# Determening how many clusters to have using within cluster sum of squares (WCSS). Set a seed to make iterations give the same result.
set.seed(123)

# determine the number of clusters using the total within sum of squares (twcss)
k_max <- 10
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})

# visualize the results. Two or 3 seems reasonable. For visualization purposes 2 is handier so lets go with that.
qplot(x = 1:k_max, y = twcss, geom = 'line')

# k-means clustering
km <-kmeans(boston_scaled, centers = 2)

# plot the normalized Boston dataset with clusters
pairs(boston_scaled, col = km$cluster)

# Tried to use ggpairs but that didn't manage to color with the clusters. One does get the correlation values and distributions with ggpairs though.
ggpairs(boston_scaled)

Per capita crime rate by town (crime) is correlated most strongly with accessibility to radial highways (rad, 0.63), full-value property-tax rate (tax, 0.58), and lower status of the population (lstat, 0.46). Rad and tax are very highly correlated (0.91), and rad and tax are both correlated somewhat strongly with lstat (~.45). Since the rad and tax are so highly correlated it is hard to disentangle what encourages crime in the area, good connections or wealth as measured by taxes. I would have imagined that lower status would have been negatively correlated with tax. The tax value is likely not a good indicator of the wealth in the area. I find it a bit hard to guess at what these correlations mean. One interesting value is that having higher proportion of blacks by town has a weak negative correlation with per capita crime rate.

Super bonus, 3D plots.

model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)
## [1] 404  13
dim(lda.fit$scaling)
## [1] 13  3
# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)

library(plotly)

# Plot with crime classes as color of the training data.
plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)
# Plot with clusters of k-means as color of the training data. 
# First one needs to do k-means with 4 clusters to compare the methods.
km3D <-kmeans(boston_scaled, centers = 4)

plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = km3D$cluster[ind])

Comparison of the methods using eye-balling: the high crime rate cluster and third cluster of k-means are rather similar. The k-means clustering seems to work better as far as visuals go: less intermingling. The crime classes are more intermingled even to the point where low and medium high crime rate classes are interspersed. Cluster 1 and low somewhat match, but cluster 2 and 4 and medium high and medium low clusters do not match well. At least a one run of the file. Maybe not another. Note that you might need to left-click poke at the figures to see anything.